A Wilson–Fermat Resonance Equation for Primes

We introduce a Wilson–Fermat resonance equation built from the productX (n) = ( (n − 1) ! + 1) (2^ (n−1) − 1), n ≥ 3. Instead of studying only divisibility, we decompose X (n) relative to the lattice n2Z asX (n) = Λ (n) n² + ε (n), where Λ (n) ∈ Z and ε (n) is the centered minimal additive correction. This yields an errorfunctionA (n) = |ε (n) |. For primes p, one has A (p) = 0. Computational evidence suggests that small values of A (n) encode a nontrivial resonance landscape for composite numbers. In particular, powers of twoexhibit an exact quasi-resonance: A (4) = 1, A (2ᵐ) = 1 for all m ≥ 4. This motivates the viewpoint that primality corresponds to exact Wilson–Fermat resonance, while certain composite families occupy the first nonzero resonance level.